We study the quantum phase transition of an itinerant antiferromagnet with cubic anisotropy in the presence of quenched disorder, paying particular attention to the locally ordered spatial regions that form in the Griffiths region. We derive an effective action where these rare regions are described in terms of static annealed disorder. A one-loop renormalization-group analysis of the effective action shows that for order-parameter dimensions p < 4, the rare regions destroy the conventional critical behavior, and the renormalized disorder flows to infinity. For order-parameter dimensions p > 4, the critical behavior is not influenced by the rare regions; it is described by the conventional dirty cubic fixed point. We also discuss the influence of the rare regions on the fluctuation-driven first-order transition in this system. © 2000 American Physical Society.